Turbofan Engine Sizing and Tradeoff Analysis via Signomial Programming. Martin A. York

Transcription

1 Turbofan Engine Sizing and Tradeoff Analysis via Signomial Programming by Martin A. York B.S., Massachusetts Institute of Technology, 2017 Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2017 c Massachusetts Institute of Technology All rights reserved. Author Department of Aeronautics and Astronautics May 25, 2017 Certified by Warren W. Hoburg Assistant Professor of Aeronautics and Astronautics Thesis Supervisor Accepted by Youssef M. Marzouk Associate Professor of Aeronautics and Astronautics Chair, Graduate Program Committee

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3 Turbofan Engine Sizing and Tradeoff Analysis via Signomial Programming by Martin A. York Submitted to the Department of Aeronautics and Astronautics on May 25, 2017, in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract This thesis presents a full 1D core+fan flowpath turbofan optimization model, based on first principles, and meant to be used during aircraft conceptual design optimization. The model is formulated as a signomial program, which is a type of optimization problem that can be solved locally using sequential convex optimization. Signomial programs can be solved reliably and efficiently, and are straightforward to integrate with other optimization models in an all-at-once manner. To demonstrate this, the turbofan model is integrated with a simple commercial aircraft sizing model. The turbofan model is validated against the Transport Aircraft System OPTimization turbofan model as well as two Georgia Tech Numerical Propulsion System Simulation turbofan models. Four integrated engine/aircraft parametric studies are performed, including a 2,460 variable multi-mission optimization that solves in 28 seconds. Thesis Supervisor: Warren W. Hoburg Title: Assistant Professor of Aeronautics and Astronautics 3

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5 Acknowledgments I d like to begin by thanking my family and friends for their support and comradery over the past five years. You helped me navigate the challenges of MIT while giving me some of my fondest memories. For this, I will always be grateful. This thesis would not have been possible without the support and guidance of my research advisor, Professor Woody Hoburg, the work of lead GPkit developer Ned Burnell, and Professor Mark Drela. Woody is a superb research mentor and brilliant engineer. He introduced me to geometric and signomial programming while pushing me to develop the best research product I could. Ned s continued development and support of GPkit, as well has his willingness to assist whenever a software roadblock was encountered, were instrumental to the success of this work. Professor Drela s TASOPT provided the inspiration for this work and his technical feedback over the course of the project was invaluable. I d like to give a special thanks to my undergraduate advisor, Professor Sheila Widnall, who helped guide me through my unique five year SB/SM program as well as the cadre of Air Force ROTC Detachment 365, especially Lt Col Karen Dillard (Ret.) and Capt Peterson Dela Cruz, who enabled me to stay at MIT for a fifth year as a graduate student. Finally, I would like to thank Aurora Flight Sciences for sponsoring this work. 5

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7 Contents 1 Introduction Optimization Formulation Model Architecture Solution Method Geometric Programming Signomial Programming Terminology Models GP- and SP-compatibility Static and Performance Models Model Derivation Combustor and Cooling Flow Mixing Model Area, Mass Flow, and Speed Constraints Fan and Compressor Maps Exhaust State Model Engine Weight Model Validation NPSS CFM56 Validation TASOPT Validation NPSS GE90 Validation

8 4 Optimum-Aircraft Parametric Studies Optimum-Aircraft Sensitivity to Specified Mission Range Optimum-Aircraft Sensitivity to Specified Minimum Climb Rate Full Mission Versus Cruise Only Optimization Multi-Mission Optimization Sensitivity Discussion Conclusion 59 A Diffuser, Fan, and Compressor Model 61 B Turbine Model 63 C Flight Profile and Aircraft Sizing Model 65 C.1 Weight Breakdown C.2 Aircraft Sizing C.3 General Aircraft Performance C.4 Climb C.5 Cruise C.6 Atmosphere Model D Signomial Equality Constraint Intuition 75 E Engine Boundary Layer Ingestion 77 8

9 List of Figures 1-1 Engine model architecture The non-convex signomial inequality drag constraint C D f(c L ) and GP approximations about two different points The signomial equality constraint C D = f(c L ) and its approximation TASOPT engine station numbering, which was adopted for this thesis E3 fan map with an estimated engine operating line in red. The map s design pressure ratio is E3 compressor map with an estimated engine operating line in red. The map s design pressure ratio is Monomial approximations to the E3 compressor map spine Monomial approximations to the E3 fan map spine Total fuel burn versus mission range Initial engine thrust, core and fan thrust, climb and cruise TSFC, as well as engine weight for a variety of mission ranges Initial fan and core thrust versus minimum initial climb rate Engine weight versus minimum initial climb rate Fan inlet area (A 2 ) versus minimum initial climb rate Initial climb and cruise TSFC versus minimum initial climb rate Sensitivity to fan design pressure ratio versus minimum initial climb rate

10 C-1 Xfoil NC130 airfoil drag data (dots) and a posynomial approximation of the data (solid line) for a Reynolds number of 20 million E-1 Cartoon illustrating boundary layer growth on a BLI equipped aircraft similar to the D

11 List of Tables 2.1 Assumed gas properties for each engine component Input values used in all three validation cases The number of GP solves and solution time for each validation case Input values used for CFM56 engine validation The two operating points used during CFM56 validation NPSS CFM56 validation results, expected to be similar when h f = 40.8 MJ/kg The three operating points used when validating the presented model against TASOPT Input values used for TASOPT engine validation TASOPT validation results with engine weight capped at 110% of the TASOPT value, expected to be similar at on design The two operating points used when validating the presented model against the GE90 like NPSS model Input values used for GE90 engine validation NPSS GE90 validation results, expected to be similar at both operating points Aircraft sizing and flight profile inputs Differences in engine size when accounting for the full mission profile and just cruise Differences in engine size for the presented multi-mission optimization formulation and a single 2,000 nm range mission optimization

12 4.4 Top engine design value sensitivities in the aircraft optimization example for a single 2,000 nm mission Top aircraft design and mission parameter sensitivities in the aircraft optimization example

13 Nomenclature a = speed of sound α = engine by-pass ratio A = flow area c k = constant in a monomial, posynomial, or signomial C D = aircraft drag coefficient C L = aircraft lift coefficient C p = working fluid constant pressure specific heat D = drag force η = efficiency F = total engine thrust f c = cooling flow by-pass ratio (ṁ cool /ṁ core ) f f = fuel/air ratio f o = one minus the percent of core mass flow bled for pressurization, electrical generation, etc. F 6 = core engine thrust F 8 = fan engine thrust F sp = overall specific thrust γ = ratio of working fluid specific heats G f = fan gearing ratio h, h t = static and stagnation enthalpy h f = fuel heat of combustion M engine = engine mass N f = normalized fan spool speed N 1 = normalized LPC spool speed 13

14 N 2 = normalized HPC spool speed ṁ = mass flow m = corrected mass flow M = Mach number m(u) = monomial function of u P, P t = static and stagnation pressure π ( ) = pressure ratio across component ( ) p(u) = posynomial function of u R = specific gas constant ρ = air density r uc = cooling flow velocity ratio s(u) = signomial function of u T, T t = static and stagnation temperature TSFC= thrust specific fuel consumption u = vector of all decision variables u = flow velocity V = aircraft velocity W engine = engine weight Z = total/static temperature ratio ( ) b = combustor quantity ( ) cool = cooling flow quantity ( ) core = core stream quantity ( ) d = diffuser quantity ( )...D = nominal design point quantity ( ) f = fan quantity ( ) fan = fan stream quantity ( ) fn = fan nozzle quantity ( ) HP = high pressure shaft quantity ( ) HPC = high pressure compressor quantity ( )...i = quantity at engine station i 14

15 ( ) LP = low pressure shaft quantity ( ) LPC = low pressure compressor quantity ( ) SL = sea level quantity ( ) t = stagnation quantity ( ) total = fan and core stream quantity ( ) +1 = ( ) plus one 15

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17 Acronyms BLI = boundary layer ingestion BPR = by-pass ratio FPR = fan pressure ratio GP = geometric program HP = high pressure HPC = high pressure compressor HPT = high pressure turbine LP = low pressure LPC = low pressure compressor LPT = low pressure turbine NLP = nonlinear program NPSS = Numerical Propulsion System Simulation OPR = overall pressure ratio SP = signomial program TASOPT = Transport Aircraft System OPTimization TOC = top of climb 17

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19 Chapter 1 Introduction A key goal of conceptual aircraft design is to quantify basic trade offs between competing mission requirements and between the various aircraft sub-systems. For an exhaustive study, multiple design parameter sweeps must be performed, ideally with an optimum conceptual aircraft produced for each point examined. Because typical aircraft design-parameter spaces are quite large, such trade studies demand a reliable and efficient system-level optimization method. As noted by Martins[21], there exists a need for new multidisciplinary design optimization (MDO) tools that exhibit fast convergence for medium and large scale problems. In pursuit of this goal, Hoburg et al.[15] and Kirschen et al.[19] have proposed formulating aircraft conceptual design models as geometric programs (GP) or signomial programs (SP). Geometric and signomial programs enable optimization problems with thousands of design variables to be reliably solved on laptop computers in a matter of seconds. Such speed and reliability is possible because these formulations can be solved via convex optimization (in the case of GP), or via sequential convex optimization (in the case of SP). One limitation of GP methods is that all physical model equations must be posed as either posynomial inequality constraints or monomial equality constraints, which at first seems far too restrictive. One objective of this thesis is to show that this is not necessarily the case, and that even quite complex physical models can be recast into the necessary forms. This is accomplished in two ways. First, many, but not all, expressions that arise in turbofan design are directly compatible with 19

20 GP or can be closely approximated by posynomial constraints. Second, relationships that are not directly GP-compatible are often SP-compatible. SP is a non-convex extension of GP that can be solved locally as a sequence of GPs. While SPs sacrifice guarantees of global optimality, they can be solved far more reliably than general nonlinear programs (NLPs). The specific example considered is the turbofan model in the Transport Aircraft System OPTimization (TASOPT)[10] conceptual design tool, which uses traditional optimization techniques. This model is a full 1D core+fan flowpath simulation based on first principles, which here will be recast into an SP-compatible form. This enables the construction of SP-compatible aircraft conceptual design models that address the complex design tradeoffs between engine and airframe parameters by treating the parameters as design variables. Such methods can produce much more realistic and higher-fidelity conceptual designs as starting points for subsequent preliminary and detailed design, with more reliability and much less time than would be required by the alternative MDO methods that combine traditional engine and airframe modules. The SP-compatible engine model developed here is compared against TASOPT and two Georgia Tech Numerical Propulsion System Simulation (NPSS)[17] models to demonstrate that it produces the correct results. To demonstrate its effectiveness for SP aircraft optimization, the model is integrated into a simple commercial transport aircraft sizing optimization problem. Example aircraft parametric studies are presented, including a 2,460 variable multi-mission optimization problem that solves in 28 seconds. 1.1 Optimization Formulation Model Architecture The presented engine model is formulated as a single multi-point optimization problem with no engine on/off design point distinctions. All constraints are applied at every point in the flight, and the model selects the engine which most optimally meets 20

21 Overall Engine Model Physical Engine Scalar (same for all operating points) Fan Area Map LPC/HPC Areas Maps Combustor Turbines Area Exhaust Areas Overall Engine Weight Vectorized variables (different values for different operating points) Cruise Top Of Climb Component Speeds Takeoff Component Fluid States Component Speeds Fluid States Thrust Speeds Thrust TSFC Fluid States Thrust TSFC TSFC Figure 1-1: Engine model architecture. all constraints. This, coupled with the fact SPs are solved all at once (i.e. there is no order of operations), greatly simplifies integrating the engine into a full aircraft system model. Figure 1-1 illustrates the engine model s overall architecture. It is worth noting no initial guesses are supplied to the presented model Solution Method The models in this thesis consist of sets of constraints that are compatible with SP. All SPs presented in this thesis were solved on a laptop computer using a combination of GPkit[7] and MOSEK[4]. GPkit, developed at MIT, is a python package that enables the fast and intuitive formulation of geometric (GP) and signomial programs. GPkit has a built in heuristic for solving SPs as a series of GP approximations. GPkit binds with open source and commercial interior point solvers to solve individual GPs. GPkit source code is available at and the turbofan engine model is available at 21

22 1.1.3 Geometric Programming Introduced in 1967 by Duffin et al. [12], a geometric program (GP) is a type of constrained optimization problem that becomes convex after a logarithmic change of variables. Modern interior point methods allow a typical sparse GP with tens of thousands of decision variables and tens of thousands of constraints to be solved in minutes on a desktop computer [6]. These solvers do not require an initial guess and guarantee convergence to a global optimum, assuming a feasible solution exists. If a feasible solution does not exist, the solver will return a certificate of infeasibility. These impressive properties are possible because a GP s objective and constraints consist of only monomial and posynomial functions, which can be transformed into convex functions in log space. A monomial is a function of the form m(u) = c n j=1 u a j j (1.1) where a j R, c R ++ and u j R ++. An example of a monomial is the common expression for lift, 1 2 ρv 2 C L S. In this case, u = (ρ, V, C L, S), c = 1/2, and a = (1, 2, 1, 1). A posynomial is a function of the form p(u) = K k=1 c k n j=1 u a jk j (1.2) where a jk R, c k R ++ and u j R ++. A posynomial is a sum of monomials. Therefore, all monomials are also one-term posynomials. A GP minimizes a posynomial objective function subject to monomial equality and posynomial inequality constraints. A GP written in standard form is minimize p 0 (u) subject to p i (u) 1, i = 1,..., n p, (1.3) m i (u) = 1, i = 1,..., n m 22

23 where p i are posynomial functions, m i are monomial functions, and u R n ++ are the decision variables. Once a problem has been formulated in the standard form (Equation 1.3), it can be solved efficiently Signomial Programming It is not always possible to formulate a design problem as a GP. This motivates the introduction of signomials. Signomials have the same form as posynomials s(u) = K k=1 c k n j=1 u a jk j (1.4) but the coefficients, c k R, can now be any (including non-positive) real numbers. A signomial program (SP) is a generalization of GP where the inequality constraints can be composed of signomial constraints of the form s(u) 0. The log transform of an SP is not a convex optimization problem, but it is a difference of convex optimization problem that can be written in log-space as minimize f 0 (x) subject to f i (x) g i (x) 0, i = 1,..., m (1.5) where f i and g i are convex. There are multiple algorithms that reliably solve signomial programs to local optima [5, 20]. This is done by solving a sequence of GPs, where each GP is a local approximation to the SP, until convergence occurs. It is worth noting that the introduction of even a single signomial constraint to any GP turns the GP into a SP, thus losing the guarantee of solution convergence to a global optimum. A favorable property of SP inequalities is that the feasible set of the convex approximation is always a subset of the original SP s feasible set, as depicted in Figure 1-2. This removes the need for trust regions and makes solving SPs substantially more reliable than solving general nonlinear programs. The previously presented difference of convex technique works only for signomial 23

24 CD CD C L (a) Signomial inequaltiy drag constraint C L (b) Convex approximation about C L = CD C L (c) Convex approximation about C L = Figure 1-2: The non-convex signomial inequality drag constraint C D f(c L ) and GP approximations about two different points. inequality, posynomial inequality, and monomial equality or inequality constraints. Signomial equality constraints can be approximated by monomials, as shown in Figure 1-3. Signomial equalities are the least desirable type of constraint due to the approximations involved. This work contains five signomial equality constraints. For additional details on how signomial equalities are approximated, see Opgenoord et al.[24]. For intuition on when signomial equality constraints are required, see Appendix D. 1.2 Terminology Before proceeding, it is useful to introduce some of the vocabulary used to describe this work. 24

25 Approximated optimal point after 1 GP iteration Projected optimal point after 1 GP iteration Figure 1-3: The signomial equality constraint C D = f(c L ) and its approximation Models A model is a set of GP and/or SP compatible constraints. The input to a model is the value of any fixed variables or constants appearing the model. Two models that share variables may be linked by concatenating their constraints GP- and SP-compatibility A constraint is GP-compatible if it can be written as either a monomial equality (Equation 1.1) or a posynomial inequality (Equation 1.2). A model is GP-compatible if its objective is a monomial or posynomial and all its constraints are GP compatible. A constraint is SP-compatible if it can be written as a signomial inequality (Equation 1.4) or equality. A model is SP-compatible if its objective is a monomial, posynomial, or ratio of posynomials and all its constraints can be written as either monomial equalities, posynomial inequalities, signomial inequalities, or signomial equalities. 25

26 1.2.3 Static and Performance Models The presented model is a multi-point optimization problem. To formulate the multipoint problem, two models are created for each engine component - a static and a performance model. The static model contains all variables and constraints that do not change between operating points, such as engine weight and nozzle areas. Performance models contain all constraints and variables that do change between operating points. For example, all constraints involving fluid states are contained in performance models. To simulate multiple engine operating points, the performance models are vectorized. When a model is vectorized, all the variables it contains become vectors, with each element corresponding to a different engine operating point. Figure 1-1 provides a visual representation of static and performance models. 26

27 Chapter 2 Model Derivation Constraint derivation follows the general framework of the TASOPT turbofan model[10], with minor changes to facilitate the removal on the on-design/off-design distinction. TASOPT station numbering was adopted and is presented in Figure 2-1. The model assumes a two spool engine with two compressors and two turbines. The model can support a geared fan. Values of C p and γ are assumed for each engine component and presented in Table 2.1. Isentropic relations were used to model working fluid state changes across turbomachinery components and a shaft power balance was enforced on both the low and high pressure shafts. Details of these models are discussed in Appendices A and B. Remaining submodels are described in the following sub-sections. d inl f fn ṁ fuel 0. m core lc c 2.5 hc 3 7 b 4.1 4a ht 2 ṁ core 5 6. P N 1 / G N 1 N m cool. 2 offt m offt 9 Figure 2-1: TASOPT engine station numbering, which was adopted for this thesis. lt tn 8 27

28 Table 2.1: Assumed gas properties for each engine component Engine Component C p [J/kg/K] γ Corresponding Air Temperature [K] Diffuser LPC HPC Combustor HPT LPT Core Exhaust Fan Exhaust Combustor and Cooling Flow Mixing Model The combustor and cooling flow constraints serve two purposes - to determine the fuel mass flow percentage and to account for the total pressure loss resulting from the mixing of the cooling flow and the working fluid in the main flow path. The flow mixing model is taken directly from TASOPT[10]. The fuel mass flow and T t4 are constrained via an enthalpy balance (Equation 2.1) while Equations 2.2 and 2.3 determine the remaining station 4 states. η b is the burner efficiency, a value less than one indicates a portion of injected fuel is not burned. The specified f c is the cooling flow bypass ratio whose typical values range from , with lower values indicating a higher engine technology level. C pfuel and h f are taken as constants equal to 2010 J/kg/K and MJ/kg respectively. T tf is the fuel s temperature when injected into the combustor and π b is the combustor pressure ratio. Both are user inputs. η b f f h f (1 f c )(h t4 h t3 ) + C pfuel f f (T t4 T tf ) (2.1) 28

29 h t4 = C pc T t4 (2.2) P t4 = π b P t3 (2.3) It is assumed the cooling flow is unregulated and engine pressure ratios are relatively constant so f c will not change between operating points. Further, it is assumed the cooling flow is discharged entirely over the first row of inlet guide vanes (station 4a) and mixes completely with the main flow before the first row of turbine blades (station 4.1). The first row of inlet guide vanes requires the majority of the cooling flow, justifying this assumption. The mixed out flow temperature at station 4.1 is computed with the enthalpy balance in Equation 2.4. Note this is a signomial equality. h t4.1 f f+1 = (1 f c + f f )h t4 + f c h t3 (2.4) The mixed-out state at station 4.1 is computed in terms of the temperature ratio Z4a, which is introduced for GP compatibility. Z 4a = (γ 1)(M 4a) 2 (2.5) P 4a = P t4 (Z 4a ) γ i γ i 1 (2.6) r uc. u 4a = M 4a γ4a RT t4 /Z 4a (2.7) Cooling flow velocity, u cool, is defined by the user input cooling flow velocity ratio u cool = r uc u 4a (2.8) Static pressure rise during mixing is neglected and the station 4.1 state is computed 29

30 using stagnation relations. Equation 2.10 is a signomial equality constraint. ( ) γ i Tt4.1 γ i 1 P t4.1 = P 4a T 4.1 (2.9) T 4.1 = T t u 4.1 (2.10) 2 C pc Rather than introduce a full momentum balance, this model approximates u 4.1 as the geometric average of core and cooling flow velocities. f f+1 u 4.1 = f f+1 u 4a α cool u cool (2.11) 2.2 Area, Mass Flow, and Speed Constraints Either the engine s thrust or the turbine inlet temperature must be constrained via Equation 2.12 or In a full aircraft optimization problem, F spec can be linked to thrust requirements in an aircraft performance model. When the engine model is run in isolation, F spec or T t4.1 must be specified by the user. F = F spec (2.12) T t4.1 = T t4spec (2.13) Component speed ratios are determined by the turbo-machinery maps (Section 2.3). Only the ratio of component speed to the component s nominal design speed is considered, so the nominal design speed is arbitrarily set to one. Thus, an LPC speed of N 1 = 1.1 should be thought of as an LPC speed 10 percent faster than the components nominal design speed, not a value 10 percent over max RPM. This model does not attempt to constrain actual RPM values. The fan and LPC both lie on the low pressure shaft so their speeds are correlated via Equation 2.14, which allows for a user selected gearing ratio G f. Additionally, 30

31 a maximum allowable speed is set for the fan and compressors. The max speed of 1.1 is estimated from TASOPT output. If an upper bound is not placed on speed, the optimizer will indefinitely increase component speed to drive OPR higher. When solving across an engine mission profile, the upper speed bound will only be achieved at the engine s most demanding operating point. N f = G f N 1 (2.14) N (2.15) N (2.16) Constraints on the mass flux through engine components are used to ensure each engine operating point corresponds to an engine of the same physical size. The station 5 and 7 exit states are determined using user specified nozzle pressure ratios as well as isentropic and stagnation relations. P t5 = π tn P t4.9 (2.17) P t7 = π fn P t2 (2.18) P i P 0 (2.19) ( ) γ P i Ti γ 1 = P ti T ti (2.20) ( Ti ) (M i ) 2 (2.21) T ti Equation 2.22 is a deviation from constraints in traditional engine models that 31

32 use Newton s method or a comparable iterative procedure. In many methods, M 5 and M 7 are set equal to 1 if M 6 or M 8 is respectively greater than 1 so that the exit nozzle is choked. If M 6 or M 8 is less than 1, then M 5 and M 7 are constrained to be less than 1. A switch is used to change constraints mid solve. It is not possible to switch constraints during a GP solve. Therefore, M 5 and M 7 are constrained to be less than or equal to 1, regardless of M 6 and M 8. For the mild choking typical in efficient turbofans, the effects of this reformulation are negligible, as confirmed by Section 3. M i 1 (2.22) Equations 2.23 to 2.33 set A 2, A 2.5, A 5, and A 7. Note M 2.5 is set by the user and M 2 is either linked to an aircraft performance model or set by the user. a i = γrt i (2.23) u i = a i M i (2.24) ρ i = P i RT i (2.25) In the static property calculations, the temperature ratio Z i is again introduced for GP compatibility. Z i = 1 + γ i 1 2 M i 2 (2.26) P i = P ti (Z i ) γ 1 γ (2.27) T i = T ti Z 1 i (2.28) 32

33 h i = C pi T i (2.29) In equation 2.30, the value of C pi R is precomputed and substituted into the constraint to make it GP compatible. u i = M i C pi RT i /(C pi R)) (2.30) ṁ fan = ρ 7 A 7 u 7 (2.31) ṁ core fo = ρ 5 A 5 u 5 /f f+1 (2.32) α = ṁ fan /ṁ core (2.33) Full turbine maps are not used to constrain turbine mass flow. Instead, it is assumed the entry to each turbine is always choked. This leads to two constraints, each setting the corrected mass flow at turbine entry equal to the estimated nominal value. m HPTD = m HPC f f+1 fo (P t2.5 /P t4.1 ) T t4.1 /T t2.5 (2.34) m LPTD = m LPC f f+1 fo (P t1.8 /P t4.5 ) T t4.5 /T t1.8 (2.35) The optimized nominal core mass flow is computed via equation ˆTi and ˆP i represent the estimated nominal state at engine station i. The values of ˆTt4, ˆPt2, and ˆT t2 are set by the user while all other ˆT and ˆP values are estimated using the isentropic relations, component design pressure ratios, and a shaft power balance. This process is shown in equation set Nominal mass flows are allowed to vary plus or minus 30 percent from their estimated values to account for uncertainty in the estimation process and ensure that if the nominal design condition is estimated 33

34 to occur at the aircraft s average altitude, the optimizer can place the nominal state anywhere in the flight. Optimization of the nominal state enables removal of the apriori specification of an engine on-design point. m componentd 1.3f f+1 fo ṁ cored ˆT ti /T ref /( ˆP ti /P ref ) (2.36) m componentd 0.7f f+1 m off ṁ cored ˆT ti /T ref /( ˆP ti /P ref ) ˆP i = π component ˆPi 1 ˆT ti = ˆT ti 1 (π component ) γ 1 γη i ˆT t4.5 = ˆT t4.1 ( ˆT t3 ˆT t2.5 ) (2.37) ˆT t4.9 = ˆT t4.5 ( ˆT t2.5 ˆT t2.1 ) ( ) ηiγ ˆTt4.5 γ 1 ˆπ HPT = ˆT t4.1 ˆP t4.5 = ˆπ HPT ˆPt3 2.3 Fan and Compressor Maps Fan and compressor maps are required to accurately constrain fan and compressor pressure ratios. Every engine has different compressor maps that result from detailed turbo-machinery design. The present model does not attempt to take into account factors causing variations in turbo-machinery maps. Instead, a simple compressor and fan map is assumed and applied to all engines. As argued in section 3, this is accurate enough for aircraft conceptual design optimization. GP compatible fan and compressor maps were derived from NASA s Energy Efficient Engine (E3) program[13] turbomachinery maps, which are presented in Figures 2-2 and 2-3. These are also the maps used in TASOPT. Blue curves are lines of constant component speed and red curves are the estimated engine operating line, or 34

35 spine. Each spine can be parameterized as either π = f( m) or π = f(n), where m is normalized corrected mass flow and N is component speed. The normalized corrected mass for for each components is defined below. m HPC = ṁ core T2.5 /T ref /(P t2.5 /P ref ) (2.38) m LPC = ṁ core T2 /T ref /(P t2 /P ref ) (2.39) m fan = ṁ fan T2 /T ref /(P t2 /P ref ) (2.40) A GP compatible monomial approximation to the functions π = f( m) and π = f(n) was developed with GPfit[16, 18]. The approximations for both the compressor and fan map spine fits are given by Equations 2.41 through 2.44 and plotted in Figures 2-4 and 2-5. π comp = (N) 5.66 (2.41) π comp = ( m) 1.22 (2.42) π fan = (N f ) (2.43) π fan = ( m) 1.37 (2.44) The fan map spine was only fit for speeds greater than 0.6. Single term fits are monomials that must pass through the origin, limiting their ability to capture fan trends for low speeds. During a typical flight, the low pressure spool speed (N 1 ) will rarely, if ever, drop below 0.6. The fitted map, combined with the constraint all pressure ratios are greater than 1, places an implicit lower bound on N 1 and N f which may lead to modeling inaccuracy at low throttle settings. This is acceptable 35

36 E3 Fan Map Pressure Ratio Normalized Corrected Mass Flow Figure 2-2: E3 fan map with an estimated engine operating line in red. The map s design pressure ratio is 1.7. due to the proportionally small amount of fuel burned at low throttle settings. A two term polynomial fit yields a better approximation of the fan map, but was not used because it adds an additional signomial constraint. Equations 2.45 through 2.47 are fan and compressor map approximations obtained by scaling the E3 map fits to an arbitrary design pressure ratio and constraining the pressure ratio be within 10 percent of the spine mass flow fit. This allows the operating point to move off the operating line while ensuring the operating point does not move into either the stall or surge regime. The user must specify at a minimum either fan, LPC, and HPC design pressure ratios or a maximum turbine inlet temperature (T t4.1 ). The user may specify all four values. Setting fan, LPC, and HPC design pressure ratios values scales the maps and is distinct from specifying a full engine ondesign operating point. If component design pressure ratios are left free, a maximum turbine inlet temperature must be specified so the cooling model prevents OPR from being driven to infinity. 36

37 30 E3 Compressor Map Pressure Ratio Normalized Corrected Mass Flow Figure 2-3: E3 compressor map with an estimated engine operating line in red. The map s design pressure ratio is 26. ( ) 1.7 π fan = (N f ) π ( fd ) 1.7 π fan (0.9)1.7908( m f ) 1.37 (2.45) π ( fd ) 1.7 π fan (1.1)1.7908( m f ) 1.37 π fd ( ) 26 π LPC = (N 1 ) 5.66 π ( LPCD ) 26 π LPC (0.9)25.049( m LPC ) 1.22 (2.46) π ( rmlp CD ) 26 π LPC (1.1)25.049( m LPC ) 1.22 π rmlp CD 37

38 30 25 Compressor Map Mass Flow Constraint Fitted Curve Actual Data Compressor Map Speed Constraint Fitted Curve Actual Data Pressure Ratio Pressure Ratio Normalized Corrected Mass Flow (a) Mass flow parametrization Compressor Speed (b) Speed parametrization Figure 2-4: Monomial approximations to the E3 compressor map spine. 1.8 Fan Map Mass Flow Constraint Fitted Curve Actual Data 1.8 Fan Map Speed Constraint Fitted Curve Actual Data Pressure Ratio Pressure Ratio Normalized Corrected Mass Flow (a) Mass flow parametrization Compressor Speed (b) Speed parametrization Figure 2-5: Monomial approximations to the E3 fan map spine. ( ) 26 π HPC = (N 2 ) 5.66 π ( HPCD ) 26 π HPC (0.9)25.049( m HPC ) 1.22 (2.47) π ( HPCD ) 26 π HPC (1.1)25.049( m HPC ) 1.22 π HPCD It is possible to fit a full compressor map instead of just the spine. However, there is no way to distinguish valid map points from points in the surge/stall regime. The optimizer will push the operating point towards these sections of the map, resulting in a physically invalid solution. 38

39 2.4 Exhaust State Model Thrust is determined with a momentum balance. Station 6 (core exhaust) and 8 (fan exhaust) velocities are computed by equations 2.48 to 2.52 which employ the stagnation relations and assume isentropic flow expansion. ( Pi P ti ) γ i 1 γ i = T i T ti (2.48) P i = P 0 (2.49) h ti = C pi T ti (2.50) h i = C pi T i (2.51) u i 2 + 2h i 2h ti (2.52) Fan and core thrust (F 8 and F 6 ) are computed with a momentum balance and summed to set the total thrust. F 8 /(αṁ core ) + u 0 u 8 (2.53) F 6 /( f o ṁ core ) + u 0 u 6 (2.54) F F 6 + F 8 (2.55) The specific thrust and corresponding thrust specific fuel consumption then follow. F sp = F/(a 0 α +1 ṁ core ) (2.56) 39

40 TSFC = f f g F sp a 0 α +1 (2.57) 2.5 Engine Weight In an aircraft optimization problem there is a downward pressure on engine weight. Consequently, the TASOPT[10] engine weight model can be relaxed into a posynomial inequality constraint. The TASOPT engine weight model is a fit to production engine data and does not account for the weight of a gearbox in a geared turbofan. The GP compatible engine mass constraint, taken from TASOPT[10], is presented below. ṁ total is defined by Equation M engine ( ṁ total lbm lbm π fπ LPC π HPC + (2.58) (100lbm/s)α ( ) 1.2 ) α lbm 5 ṁ core is written in the equivalent form ṁ total /(α +1 ) so an increase in either core or fan mass flow corresponds to an increase in engine weight, placing required downward pressure on both mass flows. ṁ total ṁ core + ṁ fan (2.59) 40

41 Chapter 3 Model Validation The presented model was validated against the output of a CFM56-7B27 like NPSS model, a GE90-94B like NPSS model, and TASOPT. The NPSS models were developed by Georgia Tech with publicly available data under the FAA s Environmental Design Space effort[22]. The NPSS output was provided by NASA. The TASOPT data was taken from a optimization run. The TASOPT engine output should mirror that of Georgia Tech s CFM56 like model because the CFM56-7B family powers all 737 Next Gen aircraft[1]. The intent of the validation studies was to verify the model s physics modeling, not to find the most optimal engine. Essentially, during validation the model was used for engine analysis instead of optimization. In all validation cases, BPR was constrained to be less than the validation data s max BPR. This prevents BPR from growing without bound. During validation, the objective function was the sum of all climb TSFCs plus ten times the cruise TSFC. Cruise TSFC was weighted by a factor of ten to capture the fact that a commercial aircraft spends the majority of each flight in cruise. Optimizing TSFC does not apply a downward pressure to engine weight. Thus, engine weight was capped at the simulated engine s predicted/actual engine weight. objective = TSFC climb + 10TSFC cruise (3.1) Component polytropic efficiencies, duct pressure losses, cooling flow bypass ratio, 41

42 Table 3.1: Input values used in all three validation cases. Variable Value Units T tf 435 K C p KJ/kg/K C p KJ/kg/K Table 3.2: The number of GP solves and solution time for each validation case. Validation Case Number of GP solves Solution Time [s] CFM TASOPT GE and max BPR are estimated from TASOPT/NPSS output. To mitigate errors due to the SP model s assumed gas properties, TASOPT computed turbine C p values were used in all three validation cases (NPSS computed C p was not available in the provided output). These values, along with the assumed fuel temperature, are presented in Table 3.1. Validation solution speeds are presented in Table NPSS CFM56 Validation The SP model s input values are given in Table 3.3. The SP model was constrained by two operating points, on-design and TOC, detailed in Table 3.4. The cruise and TOC operating points have similar ambient conditions and thrust requirements. The SP model should place the on-design point near the NPSS on-design point, producing little variation in predicted TSFC. Validation results are given in Table 3.5. The SP turbofan model was solved for two different h f values. Typically, the SP model has an h f of MJ/kg. However, Georgia Tech s NPSS model has an implied h f value of 40.8 MJ/kg, 5.12 percent less than SP model s value and 2 42

43 Table 3.3: Input values used for CFM56 engine validation. Variable Value Variable Value π fd α max π LPCD f c π HPCD η fan η b η LPC G f 1 η HPC f o η HPT η LPT η HP, η LP 0.97 π tn 0.98 π b 0.94 π d 0.98 π fn 0.98 W engine 23,201 N Table 3.4: The two operating points used during CFM56 validation. Flight Condition Altitude [ft] Mach Number Thrust [lbf] TOC 35, ,961.9 On-Design (cruise) 35, ,496.4 MJ/kg below the minimum h f of Jet A[23]. Solving the SP model with an h f of 40.8 MJ/kg reduces the percent error at each operating point by approximately 5 percent. The remaining error can be accounted for by variations in component maps and gas properties. 3.2 TASOPT Validation The SP turbofan model was validated against three TASOPT operating points: takeoff, TOC, and on-design. The parameters for each operating point are given in Table 3.6. The constant input values are given in Table 3.7. Limiting the SP engine weight 43

44 Table 3.5: NPSS CFM56 validation results, expected to be similar when h f = 40.8 MJ/kg. Flight Condition Predicted TSFC [1/hr] NPSS TSFC [1/hr] % Difference On Design (SP h f = MJ/kg) On Design (SP h f = 40.8 MJ/kg) Top of Climb (SP h f = MJ/kg) Top of Climb (SP h f = 40.8 MJ/kg) TASOPT On-Design (implied h f = MJ/kg) to the TASOPT engine weight results in TSFC errors of 10.6 percent, 18.0 percent, and 7.9 percent at takeoff, TOC, and the on-design point (cruise) respectively. This error results from the engine weight constraint, Equation 2.58, as well as the fan and compressor maps (Section 2.3), which set the component pressure ratios, placing an implicit upper bound on engine mass flow. At the on-design point, the SP engine has a core mass flow 15.6 percent lower than the TASOPT engine. To match the TASOPT engine s thrust, the SP engine must impart a larger velocity change to the working fluid; this increases TSFC. TASOPT[10] uses a different approximation to the E3 fan and compressor maps which allow its engines to achieve a greater mass flow for a given engine weight. If the SP engine weight is instead capped at 110 percent of the TASOPT engine weight, the mass flow discrepancy is reduced to 0.1 percent. The TSFC errors for this case are presented in in Table 3.8. Note the on-design TSFC error is now less than one percent. No matter the cap on engine weight, the greatest TSFC error occurs at the TOC condition. At TOC, the low pressure spool is at its max allowed speed of 1.1. As discussed in section 2.3, the SP model s fan map is conservative, particularly for high fan speeds. At a spool speed of 1.1 the SP model predicts a FPR of 1.75 while TASOPT has a FPR of 1.87, 6.28 percent higher. The SP model s lower FPR causes the engine to produce more core thrust, lowering efficiency and increasing TSFC. 44

45 Table 3.6: The three operating points used when validating the presented model against TASOPT. Flight Condition Altitude [ft] Mach Number Thrust [lbf] Take-Off ,350 TOC 35, ,768 On-Design (cruise) 35, ,986 Table 3.7: Input values used for TASOPT engine validation. Variable Value Variable Value π fd α max π LPCD f c π HPCD 3.75 η fan η b η LPC 0.88 G f 1 η HPC 0.87 f o η HPT η LPT η HP, η LP 0.97 π tn π b 0.94 π d π fn 0.98 W engine 35,008 N Table 3.8: TASOPT validation results with engine weight capped at 110% of the TASOPT value, expected to be similar at on design. Flight Condition Predicted TSFC [1/hr] TASOPT TSFC [1/hr] Percent Difference Takeoff Top of Climb On Design

46 Table 3.9: The two operating points used when validating the presented model against the GE90 like NPSS model. Flight Condition Altitude [ft] Mach Number Thrust [lbf] TOC 35, ,600 On-Design (cruise) 35, ,408.4 Table 3.10: Input values used for GE90 engine validation Variable Value Variable Value π fd 1.58 α max π LPCD 1.26 f c π HPCD η fan η b η LPC G f 1 η HPC f o η HPT η LPT η HP, η LP 0.97 π tn 0.98 π b 0.94 π d 0.98 π fn 0.98 W engine 77,399 N 3.3 NPSS GE90 Validation The two operating points used for GE90 validation are given in Table 3.9. Again, TOC conditions are similar to cruise conditions, so TSFC discrepancies should be small. The SP model s input values are given in Table 3.10 and results are presented in Table TSFC errors are due to assumed gas properties and variations in component maps. This validation case demonstrates that the presented model accurately scales from a CFM56 up to a GE90. 46

47 Table 3.11: NPSS GE90 validation results, expected to be similar at both operating points. Flight Condition Predicted TSFC [1/hr] NPSS TSFC [1/hr] Percent Difference On Design TOC

48 48

49 Chapter 4 Optimum-Aircraft Parametric Studies The engine model was integrated with a simplified commercial aircraft sizing model and the combined models were solved to find the aircraft/engine combination that burns the least amount of fuel. The model was solved with a variety of flight profiles as well as a varying number of missions, mission ranges, and minimum climb rates. Effects of these changes on engine sizing and parameter sensitivities are presented. The commercial aircraft sizing model is intentionally simple, capturing only general trends in aircraft sizing. A detailed description of the commercial sizing model is available in Appendix C. For the purposes of this thesis, each mission was discretized into four flight segments - two climb and two cruise. The objective is to minimize total fuel burn. Table 4.1 lists the input values given to the aircraft model. The same engine input values are used as during CFM56 validation (Table 3.3) with the exception of max BPR which is increased to , the maximum value from the takeoff, climb, and cruise segments of a TASOPT mission. The integrated engine/commercial aircraft sizing model has 628 free variables and solves in 6.78 seconds (6 GP iterations). 49

50 Table 4.1: Aircraft sizing and flight profile inputs. Variable Value Unit N eng 2 - W Smax 6,664 N/m 2 N pax e AR max Optimum-Aircraft Sensitivity to Specified Mission Range To demonstrate that the combined model captures the proper trends, it was solved for a variety of mission ranges. Each point on the following plots represents a unique aircraft/engine combination. Total fuel burn increased with range, as shown by Figure Fuel Burn vs Range Total Fuel Burn [N] Mission Range [nm] Figure 4-1: Total fuel burn versus mission range. Figure 4-2 presents plots of max engine thrust, fan and core thrust, initial climb and cruise TSFC, and engine weight versus mission range. All values remain roughly 50

51 constant across mission range. Max Engine Thrust [N] Max Engine Thrust vs Range Mission Range [nm] (a) Max engine thrust, which occurs during the initial climb segment, versus mission range. Initial Thrust [N] Initial Thrust vs Range Initial Fan Thrust Initial Core Thrust Mission Range [nm] (b) Fan and core thrust during the initial climb segment versus mission range. TSFC [1/hr] TSFC vs Range Initial Climb Initial Cruise Mission Range [nm] (c) Initial climb and cruise TSFC versus mission range Engine Weight [N] Engine Weight vs Range Mission Range [nm] (d) Engine weight versus mission range. Figure 4-2: Initial engine thrust, core and fan thrust, climb and cruise TSFC, as well as engine weight for a variety of mission ranges. 4.2 Optimum-Aircraft Sensitivity to Specified Minimum Climb Rate A minimum initial climb rate constraint was added to shift the nominal design point towards climb. The minimum climb rate was for normal operating conditions (i.e. both engines operating nominally). Increasing the minimum initial climb rate creates a need for increased thrust at low altitude, similar to adding a minimum balanced 51

52 field length requirement to an aircraft. The aircraft model was solved across a range of minimum climb rates. The initial thrust requirement on the engine was larger the higher the minimum climb rate. This is presented in Figure 4-3. The minimum climb rate constraint does not become active until the minimum climb rate exceeds 1,170 ft/min. The total thrust, fan thrust, and core thrust (also plotted in Figure 4-3) all increase in a near linear manner. Max Engine Thrust [N] Max Engine Thrust vs Minimum Initial Climb Rate Initial Thrsut [N] Initial Fan & Core Thrust vs Min. Initial Climb Rate Initial Fan Thrust Initial Core Thrust Minimum Initial Rate of Climb [ft/min] (a) Total initial thrust Minimum Initial Rate of Climb [ft/min] (b) Initial fan and core thrust Figure 4-3: Initial fan and core thrust versus minimum initial climb rate. The engine model predicts engine weight will increase with minimum rate of climb, as shown in Figure 4-4. This is the same as saying engine weight will increase with thrust, which is expected. As the engine is required to produce more thrust it gets physically larger. Figure 4-5 illustrates this with a plot of fan area versus minimum initial climb rate. Figure 4-6 presents the initial climb and cruise TSFC versus the minimum initial climb rate. For low minimum initial climb rates, the nominal design point remained at cruise and the cruise TSFC was virtually unaffected by the higher climb rate. However, as the climb rate continued to increase, the design point shifted toward climb and cruise TSFC began to increase. Essentially, the high minimum climb rate requirement is degrading cruise performance. A short balanced field length requirement would degrade the performance of a commercial aircraft in a similar way. 52